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Cross-correlation of Phase-Lagged Sine Wave

This example shows how to use the cross-correlation sequence to estimate the phase lag between two sine waves. The theoretical cross-correlation sequence of two sine waves at the same frequency also oscillates at that frequency. Because the sample cross-correlation sequence uses fewer and fewer samples at larger lags, the sample cross-correlation sequence also oscillates at the same frequency, but the amplitude decays as the lag increases.

Create two sine waves with frequencies of 2π/10 radians/sample. The starting phase of one sine wave is 0, while the starting phase of the other sine wave is -π radians. Add N(0,0.252) white noise to the sine wave with the phase lag of π radians. Set the random number generator to the default settings for reproducible results.

t = 0:99;
rng default;
x = cos(2*pi*1/10*t);
y = cos(2*pi*1/10*t-pi)+0.25*randn(size(t));

Obtain the sample cross-correlation sequence for two periods of the sine wave (10 samples). Plot the cross-correlation sequence and mark the known lag between the two sine waves (5 samples).

[xc,lags] = xcorr(y,x,20,'coeff');
stem(lags(21:end),xc(21:end),'markerfacecolor',[0 0 1]);
set(gca,'xtick',0:5:20);
hold on;
plot([5 5],[-1 1],'r','linewidth',2);

You see that the cross-correlation sequence peaks at lag 5 as expected and oscillates with a period of 10 samples.

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